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| Mirrors > Home > MPE Home > Th. List > asclfval | Structured version Visualization version GIF version | ||
| Description: Function value of the algebra scalar lifting function. (Contributed by Mario Carneiro, 8-Mar-2015.) |
| Ref | Expression |
|---|---|
| asclfval.a | ⊢ 𝐴 = (algSc‘𝑊) |
| asclfval.f | ⊢ 𝐹 = (Scalar‘𝑊) |
| asclfval.k | ⊢ 𝐾 = (Base‘𝐹) |
| asclfval.s | ⊢ · = ( ·𝑠 ‘𝑊) |
| asclfval.o | ⊢ 1 = (1r‘𝑊) |
| Ref | Expression |
|---|---|
| asclfval | ⊢ 𝐴 = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | asclfval.a | . 2 ⊢ 𝐴 = (algSc‘𝑊) | |
| 2 | fveq2 6906 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊)) | |
| 3 | asclfval.f | . . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊) | |
| 4 | 2, 3 | eqtr4di 2795 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹) |
| 5 | 4 | fveq2d 6910 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹)) |
| 6 | asclfval.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝐹) | |
| 7 | 5, 6 | eqtr4di 2795 | . . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾) |
| 8 | fveq2 6906 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊)) | |
| 9 | asclfval.s | . . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑊) | |
| 10 | 8, 9 | eqtr4di 2795 | . . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · ) |
| 11 | eqidd 2738 | . . . . . 6 ⊢ (𝑤 = 𝑊 → 𝑥 = 𝑥) | |
| 12 | fveq2 6906 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (1r‘𝑤) = (1r‘𝑊)) | |
| 13 | asclfval.o | . . . . . . 7 ⊢ 1 = (1r‘𝑊) | |
| 14 | 12, 13 | eqtr4di 2795 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (1r‘𝑤) = 1 ) |
| 15 | 10, 11, 14 | oveq123d 7452 | . . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤)) = (𝑥 · 1 )) |
| 16 | 7, 15 | mpteq12dv 5233 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤))) = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 ))) |
| 17 | df-ascl 21875 | . . . 4 ⊢ algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤)))) | |
| 18 | 16, 17, 6 | mptfvmpt 7248 | . . 3 ⊢ (𝑊 ∈ V → (algSc‘𝑊) = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 ))) |
| 19 | fvprc 6898 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (algSc‘𝑊) = ∅) | |
| 20 | mpt0 6710 | . . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (𝑥 · 1 )) = ∅ | |
| 21 | 19, 20 | eqtr4di 2795 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (algSc‘𝑊) = (𝑥 ∈ ∅ ↦ (𝑥 · 1 ))) |
| 22 | fvprc 6898 | . . . . . . . . 9 ⊢ (¬ 𝑊 ∈ V → (Scalar‘𝑊) = ∅) | |
| 23 | 3, 22 | eqtrid 2789 | . . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → 𝐹 = ∅) |
| 24 | 23 | fveq2d 6910 | . . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝐹) = (Base‘∅)) |
| 25 | base0 17252 | . . . . . . 7 ⊢ ∅ = (Base‘∅) | |
| 26 | 24, 25 | eqtr4di 2795 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → (Base‘𝐹) = ∅) |
| 27 | 6, 26 | eqtrid 2789 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → 𝐾 = ∅) |
| 28 | 27 | mpteq1d 5237 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )) = (𝑥 ∈ ∅ ↦ (𝑥 · 1 ))) |
| 29 | 21, 28 | eqtr4d 2780 | . . 3 ⊢ (¬ 𝑊 ∈ V → (algSc‘𝑊) = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 ))) |
| 30 | 18, 29 | pm2.61i 182 | . 2 ⊢ (algSc‘𝑊) = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )) |
| 31 | 1, 30 | eqtri 2765 | 1 ⊢ 𝐴 = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 1rcur 20178 algSccascl 21872 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addcl 11215 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-slot 17219 df-ndx 17231 df-base 17248 df-ascl 21875 |
| This theorem is referenced by: asclval 21900 asclfn 21901 asclf 21902 rnascl 21911 ressascl 21916 asclpropd 21917 rnasclg 42509 |
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